Identify the polar coordinates for each value in the table below, discuss the shape and behavior of the graphs, and use this to graph the equation of r=2 cos⁡〖3θ.〗 Symmetry hint: see if the equation is true for the following:
the line θ = 90°: replace (r, θ) with (r, 180° – θ) or (-r, -θ)
the polar axis: replace (r, θ) with (-r, 180° – θ) or (r, -θ)
the pole: replace (r, θ) with (r, 180° + θ) or (-r, θ)

we know that

the relationship between the 2-dimensional polar and Cartesian coordinates is

r = √(x² + y²)

Θ = tan⁻¹ (y/x)

so

Part a) (2, −2)> this point belong to the IV quadrant

r = √(x² + y²) r = √(2² + (-2)²)> r=√8

Θ = tan⁻¹ (y/x) Θ = tan⁻¹ (2/2)> 45°

remember that the point belong to the IV quadrant

so

Θ=360-45> Θ=315°

the answer part A) is 

(r,Θ)=(√8,315°)

Part b) (-1, 3)> this point belong to the II quadrant

r = √(x² + y²) r = √(-1² + (3)²)> r=√10

Θ = tan⁻¹ (y/x) Θ = tan⁻¹ (3/1)> 71.57°

remember that the point belong to the II quadrant

so

Θ=180-71.57> Θ=108.43°

the answer part B) is 

(r,Θ)=(√10,108.43°)

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